Let and be distinct points in a Euclidean metric space. The set is convex. Does the same apply for any linear combination of the distances, i.e.: Is , where , in any case convex?

February 11th 2010, 04:29 PM

Drexel28

Quote:

Originally Posted by Richard

Let and be distinct points in a Euclidean metric space. The set is convex. Does the same apply for any linear combination of the distances, i.e.: Is , where , in any case convex?

I just did this recently on another site :). What have you tried?

February 11th 2010, 04:40 PM

Richard

Convexity

It's clear to me that for the simple case that , the set must be convex. To generate the said set, you just need to draw a line through the two points, half it, and draw a perpendicular line through the half. The halfplane on the side of is then the said set, which is convex. But I did not manage to generalise the proof.